# Hausdorff distance and Hausdorff dimension

Let $$X$$ be a metric space, $$A$$, $$B$$ two compact subsets of $$X$$ such that the Hausdorff distance $$dist_H$$ between $$A$$ and $$B$$ is small: $$dist_H(A, B) \leq \epsilon,$$ with $$\epsilon > 0$$. Does it imply that the Hausdorff dimensions $$dim_H A$$ and $$dim_H B$$ are closed to each other ? In other words, is the Hausdorff dimension continuous with respect to the Hausdorff distance ?

Simple example. Let $$n$$ be a positive integer. Let $$A = [0,1]\quad\text{and}\quad B_n = \left\{\frac{k}{n}\;:\;k=0,1,2,\dots,n\right\}$$ Then $$\operatorname{dist}_\mathrm{H}(A,B_n) =\frac{1}{2n}$$; $$\dim_H A = 1$$; $$\dim_H B_n = 0$$.

So $$\operatorname{dist}_\mathrm{H}(A,B_n) \to 0$$ but $$\dim_H B_n \not\to \dim_H A$$.

• Thank you for this nice example. – user776060 Apr 20 at 15:15

No. For instance, if $$(X,d)$$ is locally compact and $$K\subseteq X$$ is compact, then for all $$\varepsilon>0$$ there is some relatively compact open set $$U$$ such that $$K\subseteq U\subseteq K_\varepsilon$$, and therefore $$d_H\left(K,\overline U\right)\le \varepsilon$$. However, $$\dim_H\overline U$$ is larger than the Hausdorff dimension of some open ball. And, say, in $$\Bbb R^n$$ the dimension all balls is $$n$$, while there are of course subsets of Hausdorff dimension smaller than $$1$$.

• Thank you very much for your answers. – user776060 Apr 20 at 14:28
• What if $A$ and $B$ are connected sets? – user776060 Apr 20 at 14:28
• @bill Irrelevant. In fact, if $K$ is connected and balls are connected, then $U$ may be chosen to be connected. – Gae. S. Apr 20 at 14:35
• Thank you again. However, we have $\vert diam(A) - diam(B) \vert \leq dist_H(A, B)$. So, are there any other measures or dimensions of $A$ and $B$ whose difference is bounded by the Hausdorff distance? Maybe the topological dimension? – user776060 Apr 20 at 14:45
• @bill Topological dimension won't do, because it's still $n$ for the $n$-ball. Others I don't know (which doesn't by any means exclude the possibility). – Gae. S. Apr 20 at 14:50